Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?
10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?
Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?
Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?
What is the best way to shunt these carriages so that each train can continue its journey?
How will you go about finding all the jigsaw pieces that have one peg and one hole?
Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?
In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?
A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?
Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?
Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?
Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?
Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.
The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?
Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
These practical challenges are all about making a 'tray' and covering it with paper.
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .
How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.
How many different triangles can you make on a circular pegboard that has nine pegs?
Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?
When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'.
In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?
How long does it take to brush your teeth? Can you find the matching length of time?
Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?
Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.
What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?
Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?
When you throw two regular, six-faced dice you have more chance of getting one particular result than any other. What result would that be? Why is this?
Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?
Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.
If these elves wear a different outfit every day for as many days as possible, how many days can their fun last?
Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.
There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?
A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?
Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
Place six toy ladybirds into the box so that there are two ladybirds in every column and every row.
This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?
The Vikings communicated in writing by making simple scratches on wood or stones called runes. Can you work out how their code works using the table of the alphabet?
A few extra challenges set by some young NRICH members.
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.